1. Fourier / Wavelet Analysis
ASTR 3010
Lecture 19
Textbook : N/A

2. Fourier Transform
in signal processing, (time and frequency)

3. Add bunch of zeros in your data!
Number of input data points  number of frequency sampling in FT!

4. Example of FFT in astronomy : defringing a spectrum
heavily fringed raw spectrum
power spectrum of the input
defringed spectrum

5. Limits on Fourier Transform
it can only “see” one variable (period or time) at a time at sufficient precision!

6. Short-Time Fourier Transform
Using a window function in time
Limited by the Uncertainty Principle : t*ω = constant

7. STFT resolution problem
Four different Gaussian windows

8. Wavelet Transform
Wavelet transform can get two different information (i.e., time and frequency) simultaneously!

9. Wavelet Transform where basis function is s : scale parameter
τ : translation parameter

10. Practical use of wavelet transformation
Decomposition and recomposition of a signal

11. PyWavelets pywt pywt.wavelist pywt.wavelet pywt.wavedec pywt.waverec
['bior1.1', 'bior1.3', 'bior1.5', 'bior2.2', 'bior2.4', … 'coif1', 'coif2', 'db1', 'db2', 'db3', 'sym15', 'sym16', 'sym17', 'sym18', 'sym19', 'sym20']
pywt
pywt.wavelist
pywt.wavelet
pywt.wavedec
pywt.waverec
import pywt
pywt.wavelist()

12. PyWavelets http://www.pybytes.com/pywavelets pywt pywt.wavelist
pywt.wavelet
pywt.wavedec
pywt.waverec
import pywt
myw=pywt.wavelet(‘db4’)
phi,psi,wx = myw.wavefun()
plot(wx,phi,’r’)
plot(wx,psi,’b’)
Daubechies Wavelet : order 4

13. PyWavelets http://www.pybytes.com/pywavelets pywt pywt.wavelist
pywt.wavelet
pywt.wavedec
pywt.waverec
import pywt
myw=pywt.wavelet(‘sym20’)
phi,psi,wx = myw.wavefun()
plot(wx,phi,’r’)
plot(wx,psi,’b’)

14. Wavelets Decomposition Tree
decomposition of a signal into several resolution levels.
First, the original signal is decomposed by two complementary half-band filters (high-pass and low-pass filters) that divide a spectrum into high-frequency (detail coefficients; D1) and low-frequency (approximation coefficients; A1) components (bands). For example, the low-pass filter will remove all half-band highest frequencies. Information from only the low frequency band (A1), with a half number of points, will be filtered in the second decomposition level. The A2 outcome will be filtered again for further decomposition.

15. PyWavelets decomposition reconstruction
pywt
pywt.wavelist
pywt.wavelet
pywt.wavedec
pywt.waverec
import pywt
myw=pywt.wavelet(‘db4’)
dec = myw.wavedec(data,’db4’,’zpd’,5)

16. PyWavelets decomposition reconstruction
pywt
pywt.wavelist
pywt.wavelet
pywt.wavedec
pywt.waverec
import pywt
myw=pywt.wavelet(‘sym20’)
dec = myw.wavedec(data,’sym20’,’zpd’,5)

17. pywt : Denoising
import pywt … set high order “difference” coeffs to zero. … among “diff” coeffs, clip small coeffs < 0.2*sigma … then, reconstruct dec = myw.wavedec(data,’db4’,’zpd’,5)

18. Wavelet: Denoising

19. Wavelet: Denoise in 2D

20. Wavelet: Denoise in 2D